|Considering the troubleshooting steps you've tried, the most likely suspect for being the root of this problem is the video itself.|
Since most laptops have the video card built onto the motherboard, you're looking at replacing the motherboard. In every case I've run into this situation (and I've seen this many times as I've worked on numerous laptops) a replacement motherboard is so expensive that it's not worth it to buy one. Typically they run around $1,000.00, or did 7+ years back when I last worked on a laptop.
Almost every single time I've run into a motherboard replacement situation with regard to laptops, the owners opted for just buying a new laptop.
If you want, you can take your unit into a qualified professional and have it assessed but I'd be willing to bet cash money it's the video.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.